Research
My research interests include computational fluid dynamics, shock hydrodynamics, solid mechanics, fluid-structure interaction, parallel algorithms, and numerical methods for solving partial differential equations.
Current Research Activities
- Multimaterial Methods for Shock Hydrodynamics
- Multifluid Incompressible Flows for Manufacturing Simulation
- High-Order Interface Reconstruction Methods
- Invariant Preserving Discretizations in Computational Mechanics
Some Recent and Past Research Efforts
- Methods for Coupled Fluid-Solid Interaction with Shock Loading
- Multiscale Lagrangian Hydrodynamics Methods
- Advanced Large Eddy Simulation Algorithms
- Wavelet Bases for Multiscale, Grid-Based Simulation
- DYNA for Fluids
- Parallel Run-Time Visualization for Unstructured Grid Applications
Multimaterial Methods for Shock Hydrodynamics
The use of the so-called ``uniform-strain'' or ``mean-strain'' formulation for multimaterial hydrodynamic flows has found relatively wide acceptance in the Laboratory/Engineering communities. Virtually all codes based on the mean-strain formulation rely on a Lagrangian step, some form of volume-tracking and concomitant remap procedure. The issues surrounding the so-called mean-strain formulation, pressure and pressure-temperature equilibrium, and strain-partitioning are being investigated. A hybrid multimaterial/multispecies approach based on the mean-strain formulation has been developed for problems that involve the transport of multiple chemical species in a multimaterial setting. Both direct-Eulerian and Lagrangian-remap techniques are being explored to assess the limitations of the mean-strain, pressure-equilibrium, pressure-relaxation and pressure-temperature models.
Multifluid Incompressible Flows for Manufacturing Simulation
This effort is aimed at improving the underlying flow simulation capabilities for casting problems being addressed by the Telluride Project. The flow solution algorithm is based on a second-order incremental projection method that has been extended to incorporate volume-tracking for treating multiple fluids with sharp interfaces between them. In the context of casting, the flow solution algorithm is intimately coupled to heat-transfer/phase-change to permit the simulation of metal solidification during the casting process. Currently, efforts are focused on the development of generalized edge-based advection methods that deliver second-order accuracy (on smooth data), and can be easily integrated with volume-tracking, i.e., interface reconstruction methods. In addition, efforts are underway to improve the discrete representation of the pressure-Poisson operator for multifluid problems where accuracy can be limited by high density differences between fluids.
High-Order Interface Reconstruction
The idea of interface reconstruction based on a scalar field of material volume fractions was introduced nearly 30 years ago. Since that time interface reconstruction, or more precisely, volume-tracking using interface reconstruction has been applied to a broad and diverse set of applications with sharp interfaces involving multifluid/multimaterial compressible and incompressible flows. These problems range from mold-filling to welding, tank sloshing, wave-breaking in fluid-structure problems, multimaterial forming problems, etc. The problems associated with these early methods have led to new ideas in reconstruction methods. One of the most important advances has been the introduction of linearity preserving piecewise-linear interface approximations to achieve second-order spatial accuracy. Additional ideas in this area led to the development of methods for computing the exact intersection of piecewise planar interfaces with general hexahedral elements. More recent developments have focused on the development of moment-of-fluid (MoF) methods with characteristic-based advection. This work is focused on the integration of the MoF reconstruction and characteristic-based advection in the context of a second-order projection method for multifluid flows. Preliminary results have indicated that both the flow solver and MoF yield second-order spatial accuracy.
Invariant Preserving Discretizations in Computational Mechanics
The discrete nature of shock hydro methods admits the presence of unresolved data that may be introduced in boundary or initial conditions, generated due to mismatched material impedances at interfaces, or result from nonlinear processes. Unresolved data may appear in both Lagrangian and Eulerian hydro methods, is typically revealed as spurious oscillations at the grid Nyquist limit, and corresponds to the presence of modes in the null-space in one or more discrete operators. In Lagrangian calculations, these modes can degrade accuracy, and become so severe as to become unstable and destroy the integrity of a simulation. In ALE calculations, these modes can result in significant mesh tangling defeating the most robust rezone algorithms, and degrading the accuracy of remap procedures. The relationship between discrete operators, physics, and governing balance equations makes detection and elimination of unresolved modes a very delicate issue as frequently it is difficult to distinguish the pathological from the physical modes.
In this research, we are investigating a physically-motivated and mathematically rigorous approach to modal decomposition that can be used to construct discretizations that permit a proper segregation of resolved and unresolved modes in vector fields. The use of a vector modal decomposition based on physically-relevant operators, e.g., the div and curl will overcome difficulties that have persisted for over 40 years and offers the following advantages over existing methods
Legitimately segregate resolved modes into physically-relevant components, while identifying unresolved modes
Frame-invariant and spatially local methods to control global spurious modes
Tools to identify energy components contained in spurious modes and properly filter or dissipate energy over physical time scales
Decomposition methods that may be used as rezone indicators in ALE, e.g., based on physical quantities such as the vorticity (curl of velocity)
New modal decompositions of the displacement fields for mesh rezoning in ALE that can control specific mesh features
Preliminary research results have already shown that it is possible to construct an orthogonal basis so that the decomposition a) respects both the fundamental conservation principles, e.g., conservation of mass, momentum and energy, b) is frame-invariant, and c) preserves physically relevant quantities, such as the divergence (div) and vorticity (curl of the velocity). This approach is unique because it is applicable to a broad variety of methods, e.g., compatible finite volume, finite element and particle methods (SPH, MLSPH, etc.), and it can be applied to filtering unresolved modes, ALE rezone methods, treatment of polyhedral meshes and multifluid/multimaterial flows.
Methods for Coupled fluid-Solid Interaction with Shock Loading
This work was done with Greg Tipton who was granted his PhD with distinction from the University of New Mexico in August, 2007.
This research has focused on numerical methods for the simulation of fluid-solid interaction where the solid is loaded by a shock wave, and responds with a large bulk motion. Here, a discontinuous Galerkin solver was developed to solve the compressible Euler equations on a fixed Eulerian grid using edge-based methods for unstructured and hybrid grids. A generalized level-set interface to represent the coupling between the fluid and structure was used to provide provide the necessary overlap information between the fluid and solid. A super-sampled L2 projection was used to capture the “essentially” discontinuous data used to represent the interface location. This resulted in a method that could accurately represent the interface location in a single cell. Numerical fluxes at the interface were computed using a ghost-fluid method. The sequence of images below shows a projectile (blue square) ejected from a shock tube (red) at Mach 4 and traveling through a fluid initially at rest. Additional details may be found in Greg's PhD dissertation [1]
[1] D. Gregory Tipton, “Coupled Fluid-Solid Interactions under Shock Wave Loading,” Ph.D. Dissertation, University of New Mexico, Albuquerque, New Mexico, August, 2007.
Multiscale Lagrangian Hydrodynamics Methods
Lagrangian hydro methods have been used heavily for a broad range of applications since their inception at Los Alamos during the Manhattan project. Lagrangian hydro methods form the basis for many codes in the DOE complex and in industry – both in pure Lagrangian form and in arbitrary-Lagrangian-Eulerian (ALE) form. The modern, multi-dimensional Lagrangian hydro methods that are found in widespread use today exhibit multiple numerical pathologies. These pathologies stem from ad-hoc artificial viscosities, unconstrained hourglass modes, spurious pressure-gradient approximations on distorted grids, and hinge on the use of inherently low-order staggered-grid discretizations.
Over the past several years, a collaborative effort has been underway to develop a new variational multiscale (VM) Lagrangian hydro/solid-dynamics method that permits the use of quadrilateral (hexahedral) and triangular (tetrahedral) elements, eliminates the need for explicit hourglass stabilization, and uses residual-based shock-capturing ideas rather than an ad-hoc artificial viscosity. The new VM hydro method admits implicit/explicit time-integration making it ideal for h-adaptivity, and it avoids the accuracy issues associated with currently used “slaved” time integration methods found in many existing hydrocodes today. The on-going work has built on the preliminary work carried out by Scovazzi, Christon and Hughes (see Scovazzi, Ph.D. thesis [1]) for a one-dimensional VM Lagrangian shock hydrodynamics method and has yielded a two-dimensional capability that is currently being tested/verified.
The VM hydro method is based on a Lagrangian conservation law formulation and permits the use of collocated variables, e.g., equal-order interpolation for velocities, displacements, and thermodynamic variables. This aspect makes the current formulation insensitive to the typical pathologies affecting standard Lagrangian hydrocodes. Currently, the multiscale formulation uses a residual-based discontinuity capturing operator and assumes a form substantially different from the previously designed artificial viscosity operators for stabilized finite element methods. Both modifications are due to the highly unsteady nature of multidimensional shock problems. The current VM hydro formulation is compatible with the edge-based conservative advection methods developed by Christon [2] for unstructured grids permitting a natural path to an overall ALE methodology.
[1] G. Scovazzi, "Multiscale Methods in Science and Engineering," Ph.D. Dissertation, Stanford University, June, 2004.
[2] M. Christon, D. G. Tipton, "Edge-based Godunov-type schemes on unstructured grids: accuracy and cost", draft manuscript, May, 2005.
Advanced Large Eddy Simulation Algorithms for Complex Flow Physics and Geometry
The advent of Terascale computational resources is making Large Eddy Simulation (LES) a viable technology for a broad range of applications ranging from chemically reacting flows, to unstable z-pinch interfaces, and aerodynamic design for long-haul trucks. LES is a promising alternative to traditional turbulence models because there are effectively no flow-calibrated parameters, little empiricism is required for a broad spectrum of flow regimes, and transitional and time-dependent flows can be treated. LES computes a grid-resolved filtered flow field and uses a sub-grid scale model for the dissipative scales. Although LES has been demonstrated for simple geometries, the treatment of complex geometries with unstructured grids is just beginning to be explored. The interaction between dispersive errors, grid anisotropy, filters and filter scales, and sub-grid scale models has not been quantified in a rigorous way resulting in uncertainty for many LES predictions. The ongoing research effort is seeking to advance LES models and methods by quantifying the effects of filters and filter scales for unstructured, non-orthogonal grids, under-resolved flow fields, and stochastic sub-grid scale models. This effort is relying on large-scale computational capability, advanced numerical analysis techniques and controlled numerical experiments to develop an LES toolkit with models, algorithms, standard libraries for filters and derived flow statistics, and a suite of validation and verification problems.
Wavelet Bases for Multiscale, Grid-Based Simulation
The primary objective for this work was to establish the foundation for optimal wavelet formulations for grid-based simulation based upon numerical performance, computational efficiency, parallelism, and the ability to exploit the hierarchical adaptive nature of wavelets. Wavelets are a mathematical tool that dissect data, functions and differential operators into components of different frequency and wave number with a resolution matched to the scale of each component. Wavelet bases constitute a significant shift from traditional grid-based simulation methods because the multi-resolution character of wavelets enables the development of hierarchical solutions - a critical requirement for simulation based design. Compact support wavelets are localized in space permitting the solution to be refined without local or global re-meshing in order to resolve regions of high gradients without causing overshoot phenomena (ringing) in adjacent regions where the solution is smooth. This effort has begun to develop a rigorous technical framework for the evaluation and application of wavelet bases to the solution of wave propagation , advection, and diffusion problems. These physical processes are characteristic of more complicated applications such as turbulent flow where the solution accuracy and computational requirements depend directly upon the numerical errors (i .e., numerical dispersion) associated with the computational method.
DYNA for Fluids
This research focused on the development of a new complex-geometry CFD capability based on sovling numerically – and efficiently – the time-dependent incompressible Navier-Stokes equations in 3-D using a newly-derived time-accurate second-order semi-implicit projection method with a consistent mass matrix. This technique ensures the accurate tracking of shed vortices, so vital to the successful simulation of, for example , a maneuvering submarine -- an initial driving application for the Navy Systems Program. The new CFD capability will provide the capability to treat large-scale problems with complex geometry and can be easily extended to incorporate either simple or complex turbulence models. This research has been applied to a broad range of problems ranging from flow past maneuvering submarines to flow cytometry and chemically-reacting flows.
SpyLib
SpyLib is a prototype parallel rendering toolkit for unstructured-grid data that can be embedded massively parallel finite element applications to provide run-time rendering capabilities. Spyview is based upon SpyLib and is intended to be a lightweight parallel visualization tool.
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